Find the remainder when 1 +a +`a^(2) + a^(3) + cdots + a^(2019)` is divided by a -1 ?
A)2018
B)2017
C)2019
D)0

Options

A. 2018
B. 2017
C. 2019
D. 0

Answer

To find the remainder when \(1 + a + a^2 + a^3 + \cdots + a^{2019}\) is divided by \(a - 1\), we can use the Remainder Theorem. Here’s the step-by-step solution:

### Step 1: Identify the expression
We need to evaluate the expression:
\[ S = 1 + a + a^2 + a^3 + \cdots + a^{2019} \]

### Step 2: Recognize it as a geometric series
The expression \(S\) is a geometric series with the first term \(1\) and common ratio \(a\). The number of terms in this series is \(2020\) (from \(0\) to \(2019\)).

### Step 3: Use the formula for the sum of a geometric series
The sum \(S\) of a geometric series can be calculated using the formula:
\[ S = \frac{a^{n} - 1}{a - 1} \]
where \(n\) is the number of terms. In our case, \(n = 2020\):
\[ S = \frac{a^{2020} - 1}{a - 1} \]

### Step 4: Apply the Remainder Theorem
According to the Remainder Theorem, to find the remainder of \(S\) when divided by \(a - 1\), we substitute \(a = 1\) into \(S\):
\[ S(1) = 1 + 1 + 1 + \cdots + 1 \quad (\text{2020 times}) \]
This simplifies to:
\[ S(1) = 2020 \]

### Step 5: Find the remainder
Since we are interested in the remainder when \(S\) is divided by \(a - 1\), we note that the value we calculated is \(2020\). However, we need to find the remainder when \(2020\) is divided by \(a - 1\) (which is \(0\) when \(a = 1\)).

Thus, the remainder is simply the number of terms in the series minus one:
\[ \text{Remainder} = 2020 - 1 = 2019 \]

### Final Answer
The remainder when \(1 + a + a^2 + a^3 + \cdots + a^{2019}\) is divided by \(a - 1\) is:
\[ \boxed{2019} \]